A Unified SPD Token Transformer Framework for EEG Classification: Systematic Comparison of Geometric Embeddings

TL;DR

This paper provides a theoretical and empirical comparison of geometric embeddings for SPD matrices in EEG classification, introducing a unified Transformer framework that achieves state-of-the-art results.

Contribution

It offers a formal analysis linking embedding choice to optimization dynamics and introduces a unified Transformer framework for systematic comparison of SPD embeddings in EEG tasks.

Findings

Log-Euclidean Transformer achieves state-of-the-art accuracy across datasets.

BWSPD offers competitive accuracy with similar training time.

Embedding choice impacts gradient conditioning and normalization effectiveness.

Abstract

Spatial covariance matrices of EEG signals are Symmetric Positive Definite (SPD) and lie on a Riemannian manifold, yet the theoretical connection between embedding geometry and optimization dynamics remains unexplored. We provide a formal analysis linking embedding choice to gradient conditioning and numerical stability for SPD manifolds, establishing three theoretical results: (1) BWSPD's $\sqrt{\kappa}$ gradient conditioning (vs $\kappa$ for Log-Euclidean) via Daleckii-Kre\u{\i}n matrices provides better gradient conditioning on high-dimensional inputs ($d \geq 22$), with this advantage reducing on low-dimensional inputs ($d \leq 8$) where eigendecomposition overhead dominates; (2) Embedding-Space Batch Normalization (BN-Embed) approximates Riemannian normalization up to $O(\varepsilon^2)$ error, yielding $+26\%$ accuracy on 56-channel ERP data but negligible effect on 8-channel SSVEP data, matching the channel-count-dependent prediction; (3) bi-Lipschitz bounds prove BWSPD tokens preserve manifold distances with distortion governed solely by the condition ratio $\kappa$. We validate these predictions via a unified Transformer framework comparing BWSPD, Log-Euclidean, and Euclidean embeddings within identical architecture across 1,500+ runs on three EEG paradigms (motor imagery, ERP, SSVEP; 36 subjects). Our Log-Euclidean Transformer achieves state-of-the-art performance on all datasets, substantially outperforming classical Riemannian classifiers and recent SPD baselines, while BWSPD offers competitive accuracy with similar training time.